Quasi-Static Pull-in: an Instability in Electrostatic Actuators

Abstract

We identify a new instability in electrostatic actuators dubbed quasi-static pull-in. We report experimental evidence of the instability and study its characteristics in two types of micro actuators operating in ambient air. We found that the underlying mechanism is a fast-slow dynamic interaction between slowly-varying electrostatic excitation and fast resonator response that instigate large nonresonant oscillatory orbits and eventually disappears in a global Shilnikov bifurcation. Based on these findings, we formulate and present a new taxonomy of pull-in instabilities in electrostatic actuators.Nonlinearities in electrostatic Micro-Electro-Mechanical Systems (MEMS) are a rich source of interesting dynamic phenomena. Sources of nonlinearity in electrostatic MEMS include the dependence of the electrostatic force on displacement, geometric and inertial nonlinearities, nonlinear damping mechanisms, and interactions with the substrate 1 . They result in static and dynamic bifurcations, multivaluedness, and chaos. These phenomena have been exploited to design high sensitivity sensors, large amplitude actuators, mechanical memory bits, and encryption keys 2-6 .One of the most important nonlinear phenomena in electrostatic MEMS is the pull-in instability 7 where the moving structure snaps to the actuation electrode. While significant efforts have been devoted to study this phenomenon, a consistent taxonomy of its different types and underlying mechanisms is yet to emerge. We posit that a classification system based on the ratio of frequency f of the excitation to the actuator's fundamental natural frequency f n can achieve that.Static pull-in is measured using quasi-static ramp waveforms, (f/f n → 0), which minimize inertial and damping effects. As the voltage between the actuator and an electrode increases monotonically, the stable equilibrium (node) and unstable equilibrium (saddle) coincide at a saddle-node bifurcation. Beyond this point, the actuator snaps to the electrode. Krylov and Maimon 8 and Khater et al. 3 utilized ramp waveforms with frequencies of f = 1 kHz and f = 1.8 Hz, respectively, to measure static pull-in. While a step or other waveforms may also be used, the common characteristic of static pull-in is transient (non-repeatable) dynamics.A margin of stability exists around the saddle-node bifurcation. The size of this margin is proportional to the waveform rise time and the actuator quality factor 9 . A slow rise allows for accurate determination of the bifurcation point corresponding to static pull-in voltage. A fast rise instigates transients, thereby decreasing the effective static pull-in voltage. Many researchers have investigated the boundaries of this margin 8,10,11 .Dynamic pull-in is instigated by resonant waveforms where the excitation frequency and one of the natural frequencies are integer multiples or submultiples of each other (f/f n ≈ p∕q), such as the case for primary, superharmonic, and subharmonic resonances 12,13 . Dynamic pull-in occurs at lower RMS voltage than static pull-in be...

A new wide tunability MEMS based variable capacitor using two separate electro-static vertical comb drive actuators

“…The main problem in the electrostatic type variable capacitor is the pull in instability. This phenomenon limits the range of tuning [24][25][26][27][28][29]. The movable plate can only be actuated one third of the initial gap distance; beyond this value, the two plates will snap together.…”